解析数论导论

historical introduction
chapter 1 the fundamental theorem of arithmetic
　1.1 introduction
　1.2 divisibility
　1.3 greatest common divisor
　1.4 prime numbers
　1.5 the fundamental theorem of arithmetic
　1.6 the series of reciprocals of the primes
　1.7 the euclidean algorithm
　1.8 the greatest common divisor of more than two numbers
　exercises for chapter !
chapter 2 arithmetical functions and dirichlet multiplication
　2.1 introduction
　2.2 the mebius function mn)
　2.3 the euler totient function 0(n)
　2.4 a relation connecting (0 and it
　2.5 a product formula for (n)
　2.6 the dirichlet product of arithmetical functions
　2.7 dirichlet inverses and the mebius inversion formula
　2.8 the mangoidt function a(n)
　2.9 multiplicativefunctions
　2.10 multiplicative functions and dirichlet multiplication
　2.11 the inverse of a completely multiplicative function
　2.12 liouville’s function ).(.)
　2.13 the divisor functions a,(n)
　2.14 generalized convolutions
　2.15 formal power series
　2.16 the bell series of an arithmetical function
　2.17 bell series and dirichlet multiplication
　2.18 derivatives of arithmetical functions
　2.19 the selberg identity
　exercises for chapter 2
chapter 3 averages of arithmetical functions
　3.1 introduction
　3.2 the big oh notation. asymptotic equality of functions
　3.3 euler’s summation formula
　3.4 some elementary asymptotic formulas
　3.5 the average order old{n}
　3.6 the average order of the divisor functions a,(n)
　3.7 the average order of(n)
　3.8 an application to the distribution of lattice points visiblefrom the origin
　3.9 the average order of u(n) and of a(n)
　3.10 the partial sums ora dirichlet product
　3.11 applications to #(n) and a(n)
　3.12 another identity for the partial sums of a dirichletproduct
　exercises for chapter 3
chapter 4 some elementary theorems on the distribution ofprime
　numbers
　4.1 introduction
　4.2 chebyshev’s functions (x) and ,9(x)
　4.3 relations connecting (x) and ri(x)
　4.4 some equivalent forms of the prime number theorem
　4.5 inequalities for ri(n) and pn
　4.6 shapiro’s tauberian theorem
　4.7 applications of shapiro’s theorem
　4.8 an asymptotic formula for the partial sums σpsx (i/p)
　4.9 the partial sums of the m6bius function
　4.10 brief sketch of an elementary proof of the prime numbertheorem
　4.11 selberg’s asymptotic formula
　exercises for chapter 4 lot
chapter 5 congruences
　5.1 definition and basic properties of congruences
　5.2 residue classes and complete residue systems
　5.3 linear congruences
　5.4 reduced residue systems and the euler-fermat theorem il
　5.5 polynomial congruences modulo p. lagrange’s theorem
　5.6 applications of lagrange’s theorem
　5.7 simultaneous linear congruences. the chinese remainder theoreml !
　5.8 applications of the chinese remainder theorem il
　5.9 polynomial congruences with prime power moduli
　5.10 the principle of cross-classification
　5.11 a decomposition property of reduced residue systems
　exercises for chapter 5
chapter 6 finite abelian groups and their characters
　6.1 definitions
　6.2 examples of groups and subgroups
　6.3 elementary properties of groups
　6.4 construction of subgroups
　6.5 characters of finite abelian groups
　6.6 the character group
　6.7 the orthogonality relations for characters
　6.8 dirichlet characters
　6.9 sums involving dirichlet characters
　6.10 the nonvanishing of l(i, x) for real nonprincipal x l#l
　exercises for chapter 6
chapter 7 dirichlet’s theorem on primes in arithmeticprogressions
　7.1 introduction
　7.2 dirichlet’s theorem for primes of the form 4n – i and 4n +i
　7.3 the plan of the proof of dirichlet’s theorem
　7.4 proof of lemma 7.4
　7.5 proof of lemma 7.5
　7.6 proof of lemma 7.6
　7.7 proof of lemma 7.8
　7.8 proof of lemma 7.7
　7.9 distribution of primes in arithmetic progressions
　exercises for chapter 7
chapter 8 periodic arithmetical functions and gauss sums
　8.1 functions periodic modulo k
　8.2 existence of finite fourier series for periodic arithmeticalfunctions
　8.3 ramanujan’s sum and generalizations
　8.4 multiplicative properties of the sums sk(n)
　8.5 gauss sums associated with dirichlet characters
　8.6 dirichlet characters with nonvanishing gauss sums
　8.7 induced moduli and primitive characters
　8.8 further properties of induced moduli
　8.9 the conductor of a character
　8.10 primitive characters and separable gauss sums
　8.11 the finite fourier series of the dirichlet characters
　8.12 p61ya’s inequality for the partial sums of primitivecharacters
　exercises for chapter 8
chapter 9 quadratic residues and the quadratic reciprocitylaw
　9.1 quadratic residues
　9.2 legendre’s symbol and its properties
　9.3 evaluation of(- lip) and (2]p)
　9.4 gauss’ lemma
　9.5 the quadratic reciprocity law
　9.6 applications of the reciprocity law
　9.7 the jacobi symbol
　9.8 applications to diophantine equations
　9.9 gauss sums and the quadratic reciprocity law
　9.10 the reciprocity law for quadratic gauss sums
　9.11 another proof of the quadratic reciprocity law
　exercisesfor chapter 9
chapter 10 primitive roots
　10.1 the exponent ora number mod m. primitive roots
　10.2 primitive roots and reduced residue systems
　10.3 the nonexistence of primitive roots mod 2′ for a ] 3
　10.4 the existence of primitive roots mod p for odd primes p
　10.5 primitive roots and quadratic residues
　10.6 the existence of primitive roots mod p
　10.7 the existence of primitive roots mod 2p
　10.8 the nonexistence of primitive roots in the remainingcases
　10.9 the number of primitive roots mod m
　10.10 the index calculus
　10.11 primitive roots and dirichlet characters
　10.12 real-valued dirichlet characters mod p
　10.13 primitive dirichlet characters mod p
　exercises for chapter 10
chapter 11 dirichlet series and euler products
　11.1 introduction
　11.2 the half-plane of absolute convergence of a dirichletseries
　11.3 the function defined by a dirichlet series
　11.4 multiplication of dirichlet series
　11.5 euler products
　11.6 the half-plane of convergence of a dirichlet series
　11.7 analytic properties of dirichlet series
　11.8 dirichlet series with nonnegative coefficients
　11.9 dirichlet series expressed as exponentials of dirichletseries
　11.10 mean value formulas for dirichlet series
　11.11 an integral formula for the coefficients of a dirichletseries
　11.12 an integral formula for the partial sums ora dirichletseries
　exercises for chapter ii
chapter 12 the functions ζ(s) and l(s, x)
　12.1 introduction
　12.2 properties of the gamma function
　12.3 lntegrai representation for the hurwitz zeta function
　12.4 a contour integral representation for the hurwitz zetafunction
　12.5 the analytic continuation of the hurwitz zeta function
　12.6 analytic continuation of ζ(s) and l(s, x)
　12.7 hurwitz’s formula for ζ(s, a)
　12.8 the functional equation for the riemann zeta function
　12.9 a functional equation for the hurwitz zeta function
　12.10 the functional equation for l-functions
　12.11 evaluation of ζ(-n, a)
　12.12 properties of bernoulli numbers and bernoullipolynomials
　12.13 formulas for l(0, z)
　12.14 approximation of ζ(s, a) by finite sums
　12.15 inequalities for iζ(s, a)l
　12.16 inequalities for iζ(s)l and il(s, x)l
　exercises for chapter 12
chapter 13 analytic proof of the prime number theorem
　13.1 theplan of the proof
　13.2 lemmas
　13.3 a contour integral representation for ψ(x)/x2
　13.4 upper bounds for ┃ζ(s)┃and iζ'(s)[ near the line a =1
　13.5 the nonvanishing of ζ(s) on the line a =1
　13.6 inequalities for ┃1//ζ(s) and ┃ζ'(s)ζ(s)┃
　13.7 completion of the proof of the prime number theorem
　13.8 zero-free regions for ζ(s)
　13.9 the riemann hypothesis
　13.10 application to the divisor functi6n
　13.11 application to euler’s totient
　13.12 extension of pe1ya’s inequality for character sums
　exercises for chapter 13
chapter 14 partitions
　14.1 introduction
　14.2 geometric representation of partitions
　14.3 generating functions for partitions
　14.4 euler’s pentagonal-number theorem
　14.5 combinatorial proof of euler’s pentagonal-numbertheorem
　14.6 euler’s recursion formula for p(n)
　14.7 an upper bound for p(n)
　14.8 jacobi’s triple product identity
　14.9 consequences of jacobi’s identity
　14.10 logarithmic differentiation of generating functions
　14.11 the partition identities of ramanujan
　exercises for chapter 14
bibliography
index of special symbols
index